Lowis D’A. Jackson published complete and extensive tables for facilitating the use of the Ganguillet and Kutter formula (Canal and Culvert Tables, London, 1878). To lessen calculation he puts the formula in this form:—

M = n (41.6 + 0.00281/i);

v = (√ m/n) {(M + 1.811) / (M + √m)} √ (mi).

The following table gives a selection of values of M, taken from Jackson’s tables:—

i Values of M for n =
0.010 0.012 0.015 0.017 0.020 0.025 0.030
.00001 3.2260 3.8712 4.8390 5.4842 6.4520 8.0650 9.6780
.00002 1.8210 2.1852 2.7315 3.0957 3.6420 4.5525 5.4630
.00004 1.1185 1.3422 1.6777 1.9014 2.2370 2.7962 3.3555
.00006 0.8843 1.0612 1.3264 1.5033 1.7686 2.2107 2.6529
.00008 0.7672 0.9206 1.1508 1.3042 1.5344 1.9180 2.3016
.00010 0.6970 0.8364 1.0455 1.1849 1.3940 1.7425 2.0910
.00025 0.5284 0.6341 0.7926 0.8983 1.0568 1.3210 1.5852
.00050 0.4722 0.5666 0.7083 0.8027 0.9444 1.1805 1.4166
.00075 0.4535 0.5442 0.6802 0.7709 0.9070 1.1337 1.3605
.00100 0.4441 0.5329 0.6661 0.7550 0.8882 1.1102 1.3323
.00200 0.4300 0.5160 0.6450 0.7310 0.8600 1.0750 1.2900
.00300 0.4254 0.5105 0.6381 0.7232 0.8508 1.0635 1.2762

A difficulty in the use of this formula is the selection of the coefficient of roughness. The difficulty is one which no theory will overcome, because no absolute measure of the roughness of stream beds is possible. For channels lined with timber or masonry the difficulty is not so great. The constants in that case are few and sufficiently defined. But in the case of ordinary canals and rivers the case is different, the coefficients having a much greater range. For artificial canals in rammed earth or gravel n varies from 0.0163 to 0.0301. For natural channels or rivers n varies from 0.020 to 0.035.

In Jackson’s opinion even Kutter’s numerous classes of channels seem inadequately graduated, and he proposes for artificial canals the following classification:—

I.Canals in very firm gravel, in perfect ordern = 0.02
II.Canals in earth, above the average in ordern = 0.0225
III.Canals in earth, in fair ordern = 0.025
IV.Canals in earth, below the average in ordern = 0.0275
V.Canals in earth, in rather bad order, partially
  overgrown with weeds and obstructed by detritus.		n = 0.03

Ganguillet and Kutter’s formula has been considerably used partly from its adoption in calculating tables for irrigation work in India. But it is an empirical formula of an unsatisfactory form. Some engineers apparently have assumed that because it is complicated it must be more accurate than simpler formulae. Comparison with the results of gaugings shows that this is not the case. The term involving the slope was introduced to secure agreement with some early experiments on the Mississippi, and there is strong reason for doubting the accuracy of these results.

§ 100. Bazin’s New Formula.—Bazin subsequently re-examined all the trustworthy gaugings of flow in channels and proposed a modification of the original Darcy formula which appears to be more satisfactory than any hitherto suggested (Étude d’une nouvelle formule, Paris, 1898). He points out that Darcy’s original formula, which is of the form mi/v2 = α + β/m, does not agree with experiments on channels as well as with experiments on pipes. It is an objection to it that if m increases indefinitely the limit towards which mi/v2 tends is different for different values of the roughness. It would seem that if the dimensions of a canal are indefinitely increased the variation of resistance due to differing roughness should vanish. This objection is met if it is assumed that √ (mi/v2) = α + β/√ m, so that if a is a constant mi/v2 tends to the limit a when m increases. A very careful discussion of the results of gaugings shows that they can be expressed more satisfactorily by this new formula than by Ganguillet and Kutter’s. Putting the equation in the form ζv2/2g = mi, ζ = 0.002594 (1 + γ/√ m), where γ has the following values:—